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Properties

Label 2-1110-37.26-c1-0-7
Degree $2$
Conductor $1110$
Sign $0.124 - 0.992i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (0.980 − 1.69i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 1.59·11-s + (−0.499 − 0.866i)12-s + (−2.96 + 5.13i)13-s − 1.96·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.0910 + 0.157i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + (0.370 − 0.642i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.480·11-s + (−0.144 − 0.249i)12-s + (−0.822 + 1.42i)13-s − 0.524·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.0220 + 0.0382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8277126928\)
\(L(\frac12)\) \(\approx\) \(0.8277126928\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-5.19 + 3.16i)T \)
good7 \( 1 + (-0.980 + 1.69i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.59T + 11T^{2} \)
13 \( 1 + (2.96 - 5.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0910 - 0.157i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.543 - 0.941i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
41 \( 1 + (2.18 - 3.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 - 2.58T + 47T^{2} \)
53 \( 1 + (-4.74 - 8.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.04 - 8.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.33 - 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.74 - 13.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.30 - 3.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.46T + 73T^{2} \)
79 \( 1 + (6.69 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.89 - 6.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.57 + 6.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.20T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08904733199563267326448577560, −9.299796782090332013106710148829, −8.733813717187068350195003947433, −7.33406364877869126728230359184, −7.06474271347434197745379977453, −5.68513184354278787435263528654, −4.45979506206511036404202037522, −4.00197126117173347111370348104, −2.75107929322078596192741094761, −1.38824747877735582973466204655, 0.45401652504799139742782852934, 1.90529415990135227318590774736, 3.32786053617755048151170323860, 4.95411837213170872146916328998, 5.31602241284860986343495565491, 6.34444751883395133541143184907, 7.31601648826297699797440090883, 7.88033961255813404455348390641, 8.766582760802192556765986616823, 9.384680735099343697410658709894

Graph of the $Z$-function along the critical line